Chem. 31 – 9/21 Lecture Guest Lecture Dr. Roy Dixon

Announcements I Due on Wednesday –Pipet/Buret Calibration Lab Report –Format: Pipet Report Form, Buret Calibration Plot and data for these measurements (use Lab Manual pages or photocopy your lab notebook pages if neat/organized) Last Week’s Additional Problem –returned in labs –remember to put your LAB SECTION NUMBER on all assignments turned in (grading is by lab section)

Announcements II Today’s Lecture –Error and Uncertainty Finish up Gaussian Distribution Problems t and Z based Confidence Intervals Statistical Tests –Lecture will be posted under my faculty web page (but I will also send them to Dr. Miller-Schulze for his posting method)

Example Problems Text Problems 4-2 (a), (d), (e) 4-4 (a), (b) Done already

Chapter 4 – A Little More on Distributions Measurements can be a naturally varying quantity (e.g. student heights, student test scores, Hg levels in fish in a lake) Additionally, a single quantity measured multiple times typically will give different values each time (example: real distribution of measurements of mass of an ion) Note: to be considered “accurate mass”, an ion’s mass error must be less than 5 ppm (0.007 amu in above spectrum). This is only possible by averaging measurements so that the average mass meets the requirement. 2  ~ 0.2 amu x axis is mass

Chapter 4 – A Little More on Distributions Reasons for making multiple measurements: –So one has information on the variability of the measurement (e.g. can calculate the standard deviation and uncertainty) –Average values show less deviation than single measurements –Mass spectrometer example: standard deviation in single measurement ~0.1 amu, but standard deviation in 4 s averages ~0.005 amu

Chapter 4 – Calculation of Confidence Interval 1.Confidence Interval = x + uncertainty 2.Calculation of uncertainty depends on whether σ is “ well known ” 3.If  is not well known (covered later) 4.When  is well known (not in text) Value + uncertainty = Z depends on area or desired probability At Area = 0.45 (90% both sides), Z = 1.65 At Area = (95% both sides), Z = 1.96 => larger confidence interval

Chapter 4 – Calculation of Uncertainty Example: The concentration of NO 3 - in a sample is measured many times. If the mean value and standard deviation (assume as population standard deviation) are and 0.62 ppm, respectively, what would be the expected 95% confidence interval in a 4 measurement average value? (Z for 95% CI = 1.96) What is the probability that a new measurement would exceed the upper 95% confidence limit?

Chapter 4 – Calculation of Confidence Interval with  Not Known Value + uncertainty = t = Student’s t value t depends on: - the number of samples (more samples => smaller t) - the probability of including the true value (larger probability => larger t)

Chapter 4 – Calculation of Uncertainties Example Measurement of lead in drinking water sample: –values = 12.3, 9.8, 11.4, and 13.0 ppb What is the 95% confidence interval?

Chapter 4 – Ways to Reduce Uncertainty 1.Decrease standard deviation in measurements (usually requires more skill in analysis or better equipment) 2.Analyze each sample more time (this increases n and decreases t)

Overview of Statistical Tests t-Tests: Determine if a systematic error exists in a method or between methods or if a difference exists in sample sets F-Test: Determine if there is a significant difference in standard deviations in two methods (which method is more precise) Grubbs Test: Determine if a data point can be excluded on a statistical basis

Statistical Tests Possible Outcomes Outcome #1 – There is a statistically significant result (e.g. a systematic error) –this is at some probability (e.g. 95%) –can occasionally be wrong (5% of time possible if test barely valid at 95% confidence) Outcome #2 – No significant result can be detected –this doesn’t mean there is no systematic error or difference in averages –it does mean that the systematic error, if it exists, is not detectable (e.g. not observable due to larger random errors) –It is not possible to prove a null hypothesis beyond any doubt

Statistical Tests t Tests Case 1 –used to determine if there is a significant bias by measuring a test standard and determining if there is a significant difference between the known and measured concentration Case 2 –used to determine if there is a significant differences between two methods (or samples) by measuring one sample multiple time by each method (or each sample multiple times) Case 3 –used to determine if there is a significant difference between two methods (or sample sets) by measuring multiple sample once by each method (or each sample in each set once)

Case 1 t test Example A new method for determining sulfur content in kerosene was tested on a sample known to contain 0.123% S. The measured %S were: 0.112%, 0.118%, 0.115%, and 0.119% Do the data show a significant bias at a 95% confidence level? Clearly lower, but is it significant?

Case 2 t test Example A winemaker found a barrel of wine that was labeled as a merlot, but was suspected of being part of a chardonnay wine batch and was obviously mis-labeled. To see if it was part of the chardonnay batch, the mis- labeled barrel wine and the chardonnay batch were analzyed for alcohol content. The results were as follows: –Mislabeled wine: n = 6, mean = 12.61%, S = 0.52% –Chardonnay wine: n = 4, mean = 12.53%, S = 0.48% Determine if there is a statistically significant difference in the ethanol content.

Case 3 t Test Example Case 3 t Test used when multiple samples are analyzed by two different methods (only once each method) Useful for establishing if there is a constant systematic error Example: Cl - in Ohio rainwater measured by Dixon and PNL (14 samples)

Case 3 t Test Example – Data Set and Calculations Conc. of Cl - in Rainwater (Units = uM) Sample #Dixon Cl - PNL Cl Calculations Step 1 – Calculate Difference Step 2 - Calculate mean and standard deviation in differences ave d = ( )/14 ave d = 7.49 S d = 2.44 Step 3 – Calculate t value: t Calc = 11.5

Case 3 t Test Example – Rest of Calculations Step 4 – look up t Table –(t(95%, 13 degrees of freedom) = 2.17) Step 5 – Compare t Calc with t Table, draw conclusion –t Calc >> t Table so difference is significant

t- Tests Note: These (case 2 and 3) can be applied to two different senarios: –samples (e.g. sample A and sample B, do they have the same % Ca?) –methods (analysis method A vs. analysis method B)

F - Test Similar methodology as t tests but to compare standard deviations between two methods to determine if there is a statistical difference in precision between the two methods (or variability between two sample sets) As with t tests, if F Calc > F Table, difference is statistically significant S 1 > S 2

Grubbs Test Example Purpose: To determine if an “outlier” data point can be removed from a data set Data points can be removed if observations suggest systematic errors Example: Cl lab – 4 trials with values of 30.98%, 30.87%, 31.05%, and 31.00%. Student would like less variability (to get full points for precision) Data point farthest from others is most suspicious (so 30.87%) Demonstrate calculations

Dealing with Poor Quality Data If Grubbs test fails, what can be done to improve precision? –design study to reduce standard deviations (e.g. use more precise tools) –make more measurements (this may make an outlier more extreme and should decrease confidence interval)